Just like many lessons have opened throughout this course, the final unit of the year starts with another set of number sequences. I post the first the slide of the lesson notes as students arrive, and give everyone a little time to try to write a rule for each sequence. Three minutes after the bell, I post the second slide, which gives the solutions.
The sequences are all different:
My goal here is not to introduce anything new, even there are a few details here that may present a challenge for students. I don't spend too long going over this opener. There will be time for students to practice whatever they'd like in the next few weeks.
My goal is to give students a chance to start thinking about the different kinds of functions they've seen this year, and to introduce Unit 7, which will feel different than the six units that preceded it. "The last unit of the year starts today," I say. "The unit is called Functions and Modeling. The purpose of this unit is to help you review everything you've done this year, and to apply your knowledge of different kinds of functions to different modeling tasks." I point to the opener. "One kind of modeling involves taking a sequence of numbers and writing an algebraic rule to represent it, just like you've done right here. Today, you'll start a new modeling problem."
As students worked on the opener, I distributed Mastery-Based Progress Reports, just like I have every five weeks or so throughout the year. These progress reports are detailed to the Learning Target level, but they do not list all assignments. To list assignments for the entire year's work would take 5-6 pages for each student, and it's not important for everyone to be able to see that level of detail. What is important is that everyone can see their current grade on each SLT. Here's an example of what a mid-level student might see when they receive the report today.
I put up the third slide of the lesson notes to summarize what students see. By now, they know the drill, and most kids are eager to take a look at where they stand. I listen as they express the range of emotions that come from reflecting on what has, hasn't, could have, and should have happened during ninth grade.
Next, I flip to the fourth slide to review the two kinds of Student Learning Targets: Mathematical Practices that are assessed throughout the year, and Content Learning Targets specific to each unit.
I tell students to find each kind of SLT on their Progress Reports and I begin to explain how I'll structure the work of the coming weeks. Everyone will choose which content they'd like to address, and I'll be assessing MP's continuously. To see how I offer students a choice of content, check out the next section of this lesson.
For the remaining four weeks of school, students will spend the majority of their class time working on the assignments of their choice. For most students, this will involve identifying the Student Learning Targets for which they've earned the lowest grades so far, and requesting work for each.
Students will use these "Work Request Forms", which print three to a page, before I cut them up. I put a small pile of forms on each table, and I leave a large stack of them at the front of the classroom. I post the fifth slide of the lesson notes, and I describe the process for students. Students should write the full text of the learning target they'd like to work on, along with their current grade, their target grade, and a sentence or two about why they want to work on this SLT. The target grade is important, because it helps me decide how complex the work should be. In the reflection that accompanies this lesson section, I share a few examples of work request forms.
Students should only request work for Content Learning Targets (as described in the previous section of this lesson), because work on the Mathematical Practices will be incorporated into everything they do. I tell everyone that after they request work on a learning target, "It's my homework to prepare an assignment for you." The important rule is at the bottom of the fifth slide: everyone has to finish the assignment they're working on before they can get another. "I will not give you a big pile of 'make-up work,'" I explain. "If you want to improve your grade by getting work done, then you have to start by getting the first thing done. When you do, I'll give you more work, but your immediate goal should always be to finish your current work."
Creating assignments can be time-intensive on the front end, but I save everything I use, and after a few years I have a nice library of assignments to choose from. It presents a nice opportunity for me to try out assignments I've found elsewhere but never used. There are so many great resources out there, and being able to see how a task goes with a small group of students who requested a particular SLT is a nice chance to assess the assignment for eventual incorporation into my curriculum.
No matter what content students want to work on, problem solving and modeling tasks will be built into the work. This idea is outlined on the sixth slide of the lesson notes, which provides an overview of the unit. I tell students to think of this entire unit as one big project: even though everyone might choose a different assignment, everyone should think about the functions they're using, and be trying to make beautiful work.
With the first round of work requests turned in, it's time for the first assignment of this Functions and Modeling unit. As noted on the slide #7 of the lesson notes, every student must complete this assignment before they can move on to the work of their choice. In a practical sense, this gives me a little time to prepare work based on what students want. It also helps to frame the unit, by providing an example of what I mean by "Functions and Modeling".
I give that overview as I distribute Class of 2017 Fundraiser handout, and then I flip to slide #8. When I actually present this slide in class, it builds, one bullet-point at a time. As I put each detail on the screen, I say, "So far, so good?" or otherwise ask if everything makes sense. There is space on the handout for each student to summarize the problem as they understand it, so I give everyone a minute or two to finish that before moving on.
For kids, this is a compelling task with a low barrier to entry, so it's easy for everyone to get started. After I describe it, the first step is to make a table of values, which is very approachable and which students are accustomed to doing. I might lead a discussion about how to get started on the table, and this is mostly at the prompting of the kids. I try to say as little as possible, just giving kids space to confirm that what they have makes sense.
Creating a Graph
When the table is complete, students use the grid on the back of the handout to create a graph as described in task #3. Note that when graphing total revenue, R(x), as a function of the cost of a shirt, x, we're ignoring the number of t-shirts sold, which is the middle column of the table. Indeed, that's part of what I like about this task: it contains a linear modeling problem (t-shirts sold as a function of t-shirt price) and this quadratic modeling problem that students will graph now. We'll get to writing and interpreting function rules tomorrow. The final slide of the lesson notes prompts students to write rules for n(x) and R(x), but time runs out before most students get there.
For today, the big task is get the graph done. As always, this involves a brief discussion of scale, for which I try only to be a mediator and note-taker. Students are pretty solid on graphing at this point, but they'll still want to check in to make sure "they're doing it right." Some students will need to do a second draft, almost always for reasons having something to do with scaling or naming the axes. The third one is great, actually, this student really figured out how to fill the space, but I ask students like her whether it felt to easy to "count by 68's". We decide that that isn't particularly elegant here, and that nice round numbers are both easier to count by and easier on the reader. Additionally, the first time I taught this lesson, I forgot to tell students to leave a little room at the end (we're going to find the other x-intercept tomorrow). Fortunately, this student was a good sport.
Finally, as the graphs start to take shape, I'll go to each table and ask students what they see. Their surprise that these points seem to form a parabola always surprises me - but that's exactly why this lesson is important. Our quadratic functions just ended - kids took the exam the last time we met - but still students are surprised to see this idea pop up in a modeling context. This is a great opportunity to remind students that mathematical ideas do not exist in isolation, and that the end of a unit does not mean we can forget a topic.